List of integrals of inverse hyperbolic functions explained

The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals.

• In all formulas the constant is assumed to be nonzero, and denotes the constant of integration.
• For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions.
• The ISO 80000-2 standard uses the prefix "ar-" rather than "arc-" for the inverse hyperbolic functions; we do that here.

Inverse hyperbolic sine integration formulas

$$\backslash int\backslash operatorname(ax)\backslash ,dx=\; x\backslash operatorname(ax)-\backslash frac+C$$

$$\backslash int\; x\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac-\; \backslash frac+C$$

$$\backslash int\; x^2\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac+C$$

$$\backslash int\; x^m\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac\backslash int\backslash frac\backslash ,dx\backslash quad(m\backslash ne-1)$$

$$\backslash int\backslash operatorname(ax)^2\backslash ,dx=\; 2x+x\backslash operatorname(ax)^2-\; \backslash frac+C$$

$$\backslash int\backslash operatorname(ax)^n\backslash ,dx=\; x\backslash operatorname(ax)^n-\; \backslash frac+\; n(n-1)\backslash int\backslash operatorname(ax)^\backslash ,dx$$

$$\backslash int\backslash operatorname(ax)^n\backslash ,dx=\; -\backslash frac+\; \backslash frac+\; \backslash frac\backslash int\backslash operatorname(ax)^\backslash ,dx\backslash quad(n\backslash ne-1,-2)$$

Inverse hyperbolic cosine integration formulas

$$\backslash int\backslash operatorname(ax)\backslash ,dx=\; x\backslash operatorname(ax)-\; \backslash frac+C$$

$$\backslash int\; x\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac-\; \backslash frac+C$$

$$\backslash int\; x^2\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\backslash frac+C$$

$$\backslash int\; x^m\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac\backslash int\backslash frac\backslash ,dx\backslash quad(m\backslash ne-1)$$

$$\backslash int\backslash operatorname(ax)^2\backslash ,dx=\; 2x+x\backslash operatorname(ax)^2-\; \backslash frac+C$$

$$\backslash int\backslash operatorname(ax)^n\backslash ,dx=\; x\backslash operatorname(ax)^n-\; \backslash frac+\; n(n-1)\backslash int\backslash operatorname(ax)^\backslash ,dx$$

$$\backslash int\backslash operatorname(ax)^n\backslash ,dx=\; -\backslash frac+\; \backslash frac+\; \backslash frac\backslash int\backslash operatorname(ax)^\backslash ,dx\backslash quad(n\backslash ne-1,-2)$$

Inverse hyperbolic tangent integration formulas

$$\backslash int\backslash operatorname(ax)\backslash ,dx=\; x\backslash operatorname(ax)+\; \backslash frac+C$$

$$\backslash int\; x\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac+\backslash frac+C$$

$$\backslash int\; x^2\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac+\backslash frac+C$$

$$\backslash int\; x^m\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac\backslash int\backslash frac\backslash ,dx\backslash quad(m\backslash ne-1)$$

Inverse hyperbolic cotangent integration formulas

$$\backslash int\backslash operatorname(ax)\backslash ,dx=\; x\backslash operatorname(ax)+\; \backslash frac+C$$

$$\backslash int\; x\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac+\backslash frac+C$$

$$\backslash int\; x^2\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac+\backslash frac+C$$

$$\backslash int\; x^m\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac\backslash int\backslash frac\backslash ,dx\backslash quad(m\backslash ne-1)$$

Inverse hyperbolic secant integration formulas

$$\backslash int\backslash operatorname(ax)\backslash ,dx=\; x\backslash operatorname(ax)-\; \backslash frac\backslash operatorname\backslash sqrt+C$$

$$\backslash int\; x\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac\backslash sqrt+C$$

$$\backslash int\; x^2\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac\backslash operatorname\backslash sqrt-\; \backslash frac\backslash sqrt+C$$

$$\backslash int\; x^m\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac\backslash int\backslash frac\backslash ,dx\backslash quad(m\backslash ne-1)$$

Inverse hyperbolic cosecant integration formulas

$$\backslash int\backslash operatorname(ax)\backslash ,dx=\; x\backslash operatorname(ax)+\; \backslash frac\backslash operatorname\backslash sqrt+C$$

$$\backslash int\; x\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac\backslash sqrt+C$$

$$\backslash int\; x^2\backslash operatorname(ax)\backslash ,dx=\; \backslash frac-\; \backslash frac\backslash operatorname\backslash sqrt+\; \backslash frac\backslash sqrt+C$$

$$\backslash int\; x^m\backslash operatorname(ax)\backslash ,dx=\; \backslash frac+\; \backslash frac\backslash int\backslash frac\backslash ,dx\backslash quad(m\backslash ne-1)$$